Character Table for Point Group C52h

Character table for the C_52h point group

C52h E 2 C52 2 C26 2 C52^3 2 C13 2 C52^5 2 C26^3 2 C52^7 2 C13^2 2 C52^9 2 C26^5 2 C52^11 2 C13^3 2 C4 2 C26^7 2 C52^15 2 C13^4 2 C52^17 2 C26^9 2 C52^19 2 C13^5 2 C52^21 2 C26^11 2 C52^23 2 C13^6 2 C52^25 C2 i 2 S52 2 S26 2 S52^3 2 S13 2 S52^5 2 S26^3 2 S52^7 2 S13^2 2 S52^9 2 S26^5 2 S52^11 2 S13^3 2 S4 2 S26^7 2 S52^15 2 S13^4 2 S52^17 2 S26^9 2 S52^19 2 S13^5 2 S52^21 2 S26^11 2 S52^23 2 S13^6 2 S52^25 sh <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ..T ... ....T ....... ........T ........... ............T Bg 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1g * 2.00000 1.98542 1.94188 1.87003 1.77091 1.64597 1.49702 1.32625 1.13613 0.92945 0.70921 0.47863 0.24107 0.00000 -0.24107 -0.47863 -0.70921 -0.92945 -1.13613 -1.32625 -1.49702 -1.64597 -1.77091 -1.87003 -1.94188 -1.98542 -2.00000 2.00000 -1.98542 -1.94188 -1.87003 -1.77091 -1.64597 -1.49702 -1.32625 -1.13613 -0.92945 -0.70921 -0.47863 -0.24107 0.00000 0.24107 0.47863 0.70921 0.92945 1.13613 1.32625 1.49702 1.64597 1.77091 1.87003 1.94188 1.98542 -2.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 -2.00000 -1.94188 -1.77091 -1.49702 -1.13613 -0.70921 -0.24107 0.24107 0.70921 1.13613 1.49702 1.77091 1.94188 2.00000 2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 -2.00000 -1.94188 -1.77091 -1.49702 -1.13613 -0.70921 -0.24107 0.24107 0.70921 1.13613 1.49702 1.77091 1.94188 2.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g * 2.00000 1.87003 1.49702 0.92945 0.24107 -0.47863 -1.13613 -1.64597 -1.94188 -1.98542 -1.77091 -1.32625 -0.70921 0.00000 0.70921 1.32625 1.77091 1.98542 1.94188 1.64597 1.13613 0.47863 -0.24107 -0.92945 -1.49702 -1.87003 -2.00000 2.00000 -1.87003 -1.49702 -0.92945 -0.24107 0.47863 1.13613 1.64597 1.94188 1.98542 1.77091 1.32625 0.70921 0.00000 -0.70921 -1.32625 -1.77091 -1.98542 -1.94188 -1.64597 -1.13613 -0.47863 0.24107 0.92945 1.49702 1.87003 -2.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g * 2.00000 1.64597 0.70921 -0.47863 -1.49702 -1.98542 -1.77091 -0.92945 0.24107 1.32625 1.94188 1.87003 1.13613 0.00000 -1.13613 -1.87003 -1.94188 -1.32625 -0.24107 0.92945 1.77091 1.98542 1.49702 0.47863 -0.70921 -1.64597 -2.00000 2.00000 -1.64597 -0.70921 0.47863 1.49702 1.98542 1.77091 0.92945 -0.24107 -1.32625 -1.94188 -1.87003 -1.13613 0.00000 1.13613 1.87003 1.94188 1.32625 0.24107 -0.92945 -1.77091 -1.98542 -1.49702 -0.47863 0.70921 1.64597 -2.00000 ... ... ..... ....... ......... ........... ..TT......... E6g * 2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 -2.00000 -1.49702 -0.24107 1.13613 1.94188 1.77091 0.70921 -0.70921 -1.77091 -1.94188 -1.13613 0.24107 1.49702 2.00000 2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 -2.00000 -1.49702 -0.24107 1.13613 1.94188 1.77091 0.70921 -0.70921 -1.77091 -1.94188 -1.13613 0.24107 1.49702 2.00000 ... ... ..... ....... ......... ........... TT........... E7g * 2.00000 1.32625 -0.24107 -1.64597 -1.94188 -0.92945 0.70921 1.87003 1.77091 0.47863 -1.13613 -1.98542 -1.49702 0.00000 1.49702 1.98542 1.13613 -0.47863 -1.77091 -1.87003 -0.70921 0.92945 1.94188 1.64597 0.24107 -1.32625 -2.00000 2.00000 -1.32625 0.24107 1.64597 1.94188 0.92945 -0.70921 -1.87003 -1.77091 -0.47863 1.13613 1.98542 1.49702 0.00000 -1.49702 -1.98542 -1.13613 0.47863 1.77091 1.87003 0.70921 -0.92945 -1.94188 -1.64597 -0.24107 1.32625 -2.00000 ... ... ..... ....... ......... ........... ............. E8g * 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 ... ... ..... ....... ......... ........... ............. E9g * 2.00000 0.92945 -1.13613 -1.98542 -0.70921 1.32625 1.94188 0.47863 -1.49702 -1.87003 -0.24107 1.64597 1.77091 0.00000 -1.77091 -1.64597 0.24107 1.87003 1.49702 -0.47863 -1.94188 -1.32625 0.70921 1.98542 1.13613 -0.92945 -2.00000 2.00000 -0.92945 1.13613 1.98542 0.70921 -1.32625 -1.94188 -0.47863 1.49702 1.87003 0.24107 -1.64597 -1.77091 0.00000 1.77091 1.64597 -0.24107 -1.87003 -1.49702 0.47863 1.94188 1.32625 -0.70921 -1.98542 -1.13613 0.92945 -2.00000 ... ... ..... ....... ......... ........... ............. E10g * 2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 -2.00000 -0.70921 1.49702 1.77091 -0.24107 -1.94188 -1.13613 1.13613 1.94188 0.24107 -1.77091 -1.49702 0.70921 2.00000 2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 -2.00000 -0.70921 1.49702 1.77091 -0.24107 -1.94188 -1.13613 1.13613 1.94188 0.24107 -1.77091 -1.49702 0.70921 2.00000 ... ... ..... ....... ......... ........... ............. E11g * 2.00000 0.47863 -1.77091 -1.32625 1.13613 1.87003 -0.24107 -1.98542 -0.70921 1.64597 1.49702 -0.92945 -1.94188 0.00000 1.94188 0.92945 -1.49702 -1.64597 0.70921 1.98542 0.24107 -1.87003 -1.13613 1.32625 1.77091 -0.47863 -2.00000 2.00000 -0.47863 1.77091 1.32625 -1.13613 -1.87003 0.24107 1.98542 0.70921 -1.64597 -1.49702 0.92945 1.94188 0.00000 -1.94188 -0.92945 1.49702 1.64597 -0.70921 -1.98542 -0.24107 1.87003 1.13613 -1.32625 -1.77091 0.47863 -2.00000 ... ... ..... ....... ......... ........... ............. E12g * 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 ... ... ..... ....... ......... ........... ............. E13g * 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 -2.00000 ... ... ..... ....... ......... ........... ............. E14g * 2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 -2.00000 0.24107 1.94188 -0.70921 -1.77091 1.13613 1.49702 -1.49702 -1.13613 1.77091 0.70921 -1.94188 -0.24107 2.00000 2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 -2.00000 0.24107 1.94188 -0.70921 -1.77091 1.13613 1.49702 -1.49702 -1.13613 1.77091 0.70921 -1.94188 -0.24107 2.00000 ... ... ..... ....... ......... ........... ............. E15g * 2.00000 -0.47863 -1.77091 1.32625 1.13613 -1.87003 -0.24107 1.98542 -0.70921 -1.64597 1.49702 0.92945 -1.94188 0.00000 1.94188 -0.92945 -1.49702 1.64597 0.70921 -1.98542 0.24107 1.87003 -1.13613 -1.32625 1.77091 0.47863 -2.00000 2.00000 0.47863 1.77091 -1.32625 -1.13613 1.87003 0.24107 -1.98542 0.70921 1.64597 -1.49702 -0.92945 1.94188 0.00000 -1.94188 0.92945 1.49702 -1.64597 -0.70921 1.98542 -0.24107 -1.87003 1.13613 1.32625 -1.77091 -0.47863 -2.00000 ... ... ..... ....... ......... ........... ............. E16g * 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 ... ... ..... ....... ......... ........... ............. E17g * 2.00000 -0.92945 -1.13613 1.98542 -0.70921 -1.32625 1.94188 -0.47863 -1.49702 1.87003 -0.24107 -1.64597 1.77091 0.00000 -1.77091 1.64597 0.24107 -1.87003 1.49702 0.47863 -1.94188 1.32625 0.70921 -1.98542 1.13613 0.92945 -2.00000 2.00000 0.92945 1.13613 -1.98542 0.70921 1.32625 -1.94188 0.47863 1.49702 -1.87003 0.24107 1.64597 -1.77091 0.00000 1.77091 -1.64597 -0.24107 1.87003 -1.49702 -0.47863 1.94188 -1.32625 -0.70921 1.98542 -1.13613 -0.92945 -2.00000 ... ... ..... ....... ......... ........... ............. E18g * 2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 -2.00000 1.13613 0.70921 -1.94188 1.49702 0.24107 -1.77091 1.77091 -0.24107 -1.49702 1.94188 -0.70921 -1.13613 2.00000 2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 -2.00000 1.13613 0.70921 -1.94188 1.49702 0.24107 -1.77091 1.77091 -0.24107 -1.49702 1.94188 -0.70921 -1.13613 2.00000 ... ... ..... ....... ......... ........... ............. E19g * 2.00000 -1.32625 -0.24107 1.64597 -1.94188 0.92945 0.70921 -1.87003 1.77091 -0.47863 -1.13613 1.98542 -1.49702 0.00000 1.49702 -1.98542 1.13613 0.47863 -1.77091 1.87003 -0.70921 -0.92945 1.94188 -1.64597 0.24107 1.32625 -2.00000 2.00000 1.32625 0.24107 -1.64597 1.94188 -0.92945 -0.70921 1.87003 -1.77091 0.47863 1.13613 -1.98542 1.49702 0.00000 -1.49702 1.98542 -1.13613 -0.47863 1.77091 -1.87003 0.70921 0.92945 -1.94188 1.64597 -0.24107 -1.32625 -2.00000 ... ... ..... ....... ......... ........... ............. E20g * 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 ... ... ..... ....... ......... ........... ............. E21g * 2.00000 -1.64597 0.70921 0.47863 -1.49702 1.98542 -1.77091 0.92945 0.24107 -1.32625 1.94188 -1.87003 1.13613 0.00000 -1.13613 1.87003 -1.94188 1.32625 -0.24107 -0.92945 1.77091 -1.98542 1.49702 -0.47863 -0.70921 1.64597 -2.00000 2.00000 1.64597 -0.70921 -0.47863 1.49702 -1.98542 1.77091 -0.92945 -0.24107 1.32625 -1.94188 1.87003 -1.13613 0.00000 1.13613 -1.87003 1.94188 -1.32625 0.24107 0.92945 -1.77091 1.98542 -1.49702 0.47863 0.70921 -1.64597 -2.00000 ... ... ..... ....... ......... ........... ............. E22g * 2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 -2.00000 1.77091 -1.13613 0.24107 0.70921 -1.49702 1.94188 -1.94188 1.49702 -0.70921 -0.24107 1.13613 -1.77091 2.00000 2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 -2.00000 1.77091 -1.13613 0.24107 0.70921 -1.49702 1.94188 -1.94188 1.49702 -0.70921 -0.24107 1.13613 -1.77091 2.00000 ... ... ..... ....... ......... ........... ............. E23g * 2.00000 -1.87003 1.49702 -0.92945 0.24107 0.47863 -1.13613 1.64597 -1.94188 1.98542 -1.77091 1.32625 -0.70921 0.00000 0.70921 -1.32625 1.77091 -1.98542 1.94188 -1.64597 1.13613 -0.47863 -0.24107 0.92945 -1.49702 1.87003 -2.00000 2.00000 1.87003 -1.49702 0.92945 -0.24107 -0.47863 1.13613 -1.64597 1.94188 -1.98542 1.77091 -1.32625 0.70921 0.00000 -0.70921 1.32625 -1.77091 1.98542 -1.94188 1.64597 -1.13613 0.47863 0.24107 -0.92945 1.49702 -1.87003 -2.00000 ... ... ..... ....... ......... ........... ............. E24g * 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 ... ... ..... ....... ......... ........... ............. E25g * 2.00000 -1.98542 1.94188 -1.87003 1.77091 -1.64597 1.49702 -1.32625 1.13613 -0.92945 0.70921 -0.47863 0.24107 0.00000 -0.24107 0.47863 -0.70921 0.92945 -1.13613 1.32625 -1.49702 1.64597 -1.77091 1.87003 -1.94188 1.98542 -2.00000 2.00000 1.98542 -1.94188 1.87003 -1.77091 1.64597 -1.49702 1.32625 -1.13613 0.92945 -0.70921 0.47863 -0.24107 0.00000 0.24107 -0.47863 0.70921 -0.92945 1.13613 -1.32625 1.49702 -1.64597 1.77091 -1.87003 1.94188 -1.98542 -2.00000 ... ... ..... ....... ......... ........... ............. Au 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ..T ..... ......T ......... ..........T ............. Bu 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. E1u * 2.00000 1.98542 1.94188 1.87003 1.77091 1.64597 1.49702 1.32625 1.13613 0.92945 0.70921 0.47863 0.24107 0.00000 -0.24107 -0.47863 -0.70921 -0.92945 -1.13613 -1.32625 -1.49702 -1.64597 -1.77091 -1.87003 -1.94188 -1.98542 -2.00000 -2.00000 1.98542 1.94188 1.87003 1.77091 1.64597 1.49702 1.32625 1.13613 0.92945 0.70921 0.47863 0.24107 0.00000 -0.24107 -0.47863 -0.70921 -0.92945 -1.13613 -1.32625 -1.49702 -1.64597 -1.77091 -1.87003 -1.94188 -1.98542 2.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 -2.00000 -1.94188 -1.77091 -1.49702 -1.13613 -0.70921 -0.24107 0.24107 0.70921 1.13613 1.49702 1.77091 1.94188 2.00000 -2.00000 -1.94188 -1.77091 -1.49702 -1.13613 -0.70921 -0.24107 0.24107 0.70921 1.13613 1.49702 1.77091 1.94188 2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 -2.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u * 2.00000 1.87003 1.49702 0.92945 0.24107 -0.47863 -1.13613 -1.64597 -1.94188 -1.98542 -1.77091 -1.32625 -0.70921 0.00000 0.70921 1.32625 1.77091 1.98542 1.94188 1.64597 1.13613 0.47863 -0.24107 -0.92945 -1.49702 -1.87003 -2.00000 -2.00000 1.87003 1.49702 0.92945 0.24107 -0.47863 -1.13613 -1.64597 -1.94188 -1.98542 -1.77091 -1.32625 -0.70921 0.00000 0.70921 1.32625 1.77091 1.98542 1.94188 1.64597 1.13613 0.47863 -0.24107 -0.92945 -1.49702 -1.87003 2.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u * 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 -2.00000 -1.77091 -1.13613 -0.24107 0.70921 1.49702 1.94188 1.94188 1.49702 0.70921 -0.24107 -1.13613 -1.77091 -2.00000 -1.77091 -1.13613 -0.24107 0.70921 1.49702 1.94188 1.94188 1.49702 0.70921 -0.24107 -1.13613 -1.77091 -2.00000 ... ... ..... ....... ......... ..TT....... ............. E5u * 2.00000 1.64597 0.70921 -0.47863 -1.49702 -1.98542 -1.77091 -0.92945 0.24107 1.32625 1.94188 1.87003 1.13613 0.00000 -1.13613 -1.87003 -1.94188 -1.32625 -0.24107 0.92945 1.77091 1.98542 1.49702 0.47863 -0.70921 -1.64597 -2.00000 -2.00000 1.64597 0.70921 -0.47863 -1.49702 -1.98542 -1.77091 -0.92945 0.24107 1.32625 1.94188 1.87003 1.13613 0.00000 -1.13613 -1.87003 -1.94188 -1.32625 -0.24107 0.92945 1.77091 1.98542 1.49702 0.47863 -0.70921 -1.64597 2.00000 ... ... ..... ....... ......... TT......... ............. E6u * 2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 -2.00000 -1.49702 -0.24107 1.13613 1.94188 1.77091 0.70921 -0.70921 -1.77091 -1.94188 -1.13613 0.24107 1.49702 2.00000 -2.00000 -1.49702 -0.24107 1.13613 1.94188 1.77091 0.70921 -0.70921 -1.77091 -1.94188 -1.13613 0.24107 1.49702 2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 -2.00000 ... ... ..... ....... ......... ........... ............. E7u * 2.00000 1.32625 -0.24107 -1.64597 -1.94188 -0.92945 0.70921 1.87003 1.77091 0.47863 -1.13613 -1.98542 -1.49702 0.00000 1.49702 1.98542 1.13613 -0.47863 -1.77091 -1.87003 -0.70921 0.92945 1.94188 1.64597 0.24107 -1.32625 -2.00000 -2.00000 1.32625 -0.24107 -1.64597 -1.94188 -0.92945 0.70921 1.87003 1.77091 0.47863 -1.13613 -1.98542 -1.49702 0.00000 1.49702 1.98542 1.13613 -0.47863 -1.77091 -1.87003 -0.70921 0.92945 1.94188 1.64597 0.24107 -1.32625 2.00000 ... ... ..... ....... ......... ........... ............. E8u * 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 -2.00000 -1.13613 0.70921 1.94188 1.49702 -0.24107 -1.77091 -1.77091 -0.24107 1.49702 1.94188 0.70921 -1.13613 -2.00000 -1.13613 0.70921 1.94188 1.49702 -0.24107 -1.77091 -1.77091 -0.24107 1.49702 1.94188 0.70921 -1.13613 -2.00000 ... ... ..... ....... ......... ........... ............. E9u * 2.00000 0.92945 -1.13613 -1.98542 -0.70921 1.32625 1.94188 0.47863 -1.49702 -1.87003 -0.24107 1.64597 1.77091 0.00000 -1.77091 -1.64597 0.24107 1.87003 1.49702 -0.47863 -1.94188 -1.32625 0.70921 1.98542 1.13613 -0.92945 -2.00000 -2.00000 0.92945 -1.13613 -1.98542 -0.70921 1.32625 1.94188 0.47863 -1.49702 -1.87003 -0.24107 1.64597 1.77091 0.00000 -1.77091 -1.64597 0.24107 1.87003 1.49702 -0.47863 -1.94188 -1.32625 0.70921 1.98542 1.13613 -0.92945 2.00000 ... ... ..... ....... ......... ........... ............. E10u * 2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 -2.00000 -0.70921 1.49702 1.77091 -0.24107 -1.94188 -1.13613 1.13613 1.94188 0.24107 -1.77091 -1.49702 0.70921 2.00000 -2.00000 -0.70921 1.49702 1.77091 -0.24107 -1.94188 -1.13613 1.13613 1.94188 0.24107 -1.77091 -1.49702 0.70921 2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 -2.00000 ... ... ..... ....... ......... ........... ............. E11u * 2.00000 0.47863 -1.77091 -1.32625 1.13613 1.87003 -0.24107 -1.98542 -0.70921 1.64597 1.49702 -0.92945 -1.94188 0.00000 1.94188 0.92945 -1.49702 -1.64597 0.70921 1.98542 0.24107 -1.87003 -1.13613 1.32625 1.77091 -0.47863 -2.00000 -2.00000 0.47863 -1.77091 -1.32625 1.13613 1.87003 -0.24107 -1.98542 -0.70921 1.64597 1.49702 -0.92945 -1.94188 0.00000 1.94188 0.92945 -1.49702 -1.64597 0.70921 1.98542 0.24107 -1.87003 -1.13613 1.32625 1.77091 -0.47863 2.00000 ... ... ..... ....... ......... ........... ............. E12u * 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 -2.00000 -0.24107 1.94188 0.70921 -1.77091 -1.13613 1.49702 1.49702 -1.13613 -1.77091 0.70921 1.94188 -0.24107 -2.00000 -0.24107 1.94188 0.70921 -1.77091 -1.13613 1.49702 1.49702 -1.13613 -1.77091 0.70921 1.94188 -0.24107 -2.00000 ... ... ..... ....... ......... ........... ............. E13u * 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 -2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 2.00000 ... ... ..... ....... ......... ........... ............. E14u * 2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 -2.00000 0.24107 1.94188 -0.70921 -1.77091 1.13613 1.49702 -1.49702 -1.13613 1.77091 0.70921 -1.94188 -0.24107 2.00000 -2.00000 0.24107 1.94188 -0.70921 -1.77091 1.13613 1.49702 -1.49702 -1.13613 1.77091 0.70921 -1.94188 -0.24107 2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 -2.00000 ... ... ..... ....... ......... ........... ............. E15u * 2.00000 -0.47863 -1.77091 1.32625 1.13613 -1.87003 -0.24107 1.98542 -0.70921 -1.64597 1.49702 0.92945 -1.94188 0.00000 1.94188 -0.92945 -1.49702 1.64597 0.70921 -1.98542 0.24107 1.87003 -1.13613 -1.32625 1.77091 0.47863 -2.00000 -2.00000 -0.47863 -1.77091 1.32625 1.13613 -1.87003 -0.24107 1.98542 -0.70921 -1.64597 1.49702 0.92945 -1.94188 0.00000 1.94188 -0.92945 -1.49702 1.64597 0.70921 -1.98542 0.24107 1.87003 -1.13613 -1.32625 1.77091 0.47863 2.00000 ... ... ..... ....... ......... ........... ............. E16u * 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 -2.00000 0.70921 1.49702 -1.77091 -0.24107 1.94188 -1.13613 -1.13613 1.94188 -0.24107 -1.77091 1.49702 0.70921 -2.00000 0.70921 1.49702 -1.77091 -0.24107 1.94188 -1.13613 -1.13613 1.94188 -0.24107 -1.77091 1.49702 0.70921 -2.00000 ... ... ..... ....... ......... ........... ............. E17u * 2.00000 -0.92945 -1.13613 1.98542 -0.70921 -1.32625 1.94188 -0.47863 -1.49702 1.87003 -0.24107 -1.64597 1.77091 0.00000 -1.77091 1.64597 0.24107 -1.87003 1.49702 0.47863 -1.94188 1.32625 0.70921 -1.98542 1.13613 0.92945 -2.00000 -2.00000 -0.92945 -1.13613 1.98542 -0.70921 -1.32625 1.94188 -0.47863 -1.49702 1.87003 -0.24107 -1.64597 1.77091 0.00000 -1.77091 1.64597 0.24107 -1.87003 1.49702 0.47863 -1.94188 1.32625 0.70921 -1.98542 1.13613 0.92945 2.00000 ... ... ..... ....... ......... ........... ............. E18u * 2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 -2.00000 1.13613 0.70921 -1.94188 1.49702 0.24107 -1.77091 1.77091 -0.24107 -1.49702 1.94188 -0.70921 -1.13613 2.00000 -2.00000 1.13613 0.70921 -1.94188 1.49702 0.24107 -1.77091 1.77091 -0.24107 -1.49702 1.94188 -0.70921 -1.13613 2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 -2.00000 ... ... ..... ....... ......... ........... ............. E19u * 2.00000 -1.32625 -0.24107 1.64597 -1.94188 0.92945 0.70921 -1.87003 1.77091 -0.47863 -1.13613 1.98542 -1.49702 0.00000 1.49702 -1.98542 1.13613 0.47863 -1.77091 1.87003 -0.70921 -0.92945 1.94188 -1.64597 0.24107 1.32625 -2.00000 -2.00000 -1.32625 -0.24107 1.64597 -1.94188 0.92945 0.70921 -1.87003 1.77091 -0.47863 -1.13613 1.98542 -1.49702 0.00000 1.49702 -1.98542 1.13613 0.47863 -1.77091 1.87003 -0.70921 -0.92945 1.94188 -1.64597 0.24107 1.32625 2.00000 ... ... ..... ....... ......... ........... ............. E20u * 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 -2.00000 1.49702 -0.24107 -1.13613 1.94188 -1.77091 0.70921 0.70921 -1.77091 1.94188 -1.13613 -0.24107 1.49702 -2.00000 1.49702 -0.24107 -1.13613 1.94188 -1.77091 0.70921 0.70921 -1.77091 1.94188 -1.13613 -0.24107 1.49702 -2.00000 ... ... ..... ....... ......... ........... ............. E21u * 2.00000 -1.64597 0.70921 0.47863 -1.49702 1.98542 -1.77091 0.92945 0.24107 -1.32625 1.94188 -1.87003 1.13613 0.00000 -1.13613 1.87003 -1.94188 1.32625 -0.24107 -0.92945 1.77091 -1.98542 1.49702 -0.47863 -0.70921 1.64597 -2.00000 -2.00000 -1.64597 0.70921 0.47863 -1.49702 1.98542 -1.77091 0.92945 0.24107 -1.32625 1.94188 -1.87003 1.13613 0.00000 -1.13613 1.87003 -1.94188 1.32625 -0.24107 -0.92945 1.77091 -1.98542 1.49702 -0.47863 -0.70921 1.64597 2.00000 ... ... ..... ....... ......... ........... ............. E22u * 2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 -2.00000 1.77091 -1.13613 0.24107 0.70921 -1.49702 1.94188 -1.94188 1.49702 -0.70921 -0.24107 1.13613 -1.77091 2.00000 -2.00000 1.77091 -1.13613 0.24107 0.70921 -1.49702 1.94188 -1.94188 1.49702 -0.70921 -0.24107 1.13613 -1.77091 2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 -2.00000 ... ... ..... ....... ......... ........... ............. E23u * 2.00000 -1.87003 1.49702 -0.92945 0.24107 0.47863 -1.13613 1.64597 -1.94188 1.98542 -1.77091 1.32625 -0.70921 0.00000 0.70921 -1.32625 1.77091 -1.98542 1.94188 -1.64597 1.13613 -0.47863 -0.24107 0.92945 -1.49702 1.87003 -2.00000 -2.00000 -1.87003 1.49702 -0.92945 0.24107 0.47863 -1.13613 1.64597 -1.94188 1.98542 -1.77091 1.32625 -0.70921 0.00000 0.70921 -1.32625 1.77091 -1.98542 1.94188 -1.64597 1.13613 -0.47863 -0.24107 0.92945 -1.49702 1.87003 2.00000 ... ... ..... ....... ......... ........... ............. E24u * 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 -2.00000 1.94188 -1.77091 1.49702 -1.13613 0.70921 -0.24107 -0.24107 0.70921 -1.13613 1.49702 -1.77091 1.94188 -2.00000 1.94188 -1.77091 1.49702 -1.13613 0.70921 -0.24107 -0.24107 0.70921 -1.13613 1.49702 -1.77091 1.94188 -2.00000 ... ... ..... ....... ......... ........... ............. E25u * 2.00000 -1.98542 1.94188 -1.87003 1.77091 -1.64597 1.49702 -1.32625 1.13613 -0.92945 0.70921 -0.47863 0.24107 0.00000 -0.24107 0.47863 -0.70921 0.92945 -1.13613 1.32625 -1.49702 1.64597 -1.77091 1.87003 -1.94188 1.98542 -2.00000 -2.00000 -1.98542 1.94188 -1.87003 1.77091 -1.64597 1.49702 -1.32625 1.13613 -0.92945 0.70921 -0.47863 0.24107 0.00000 -0.24107 0.47863 -0.70921 0.92945 -1.13613 1.32625 -1.49702 1.64597 -1.77091 1.87003 -1.94188 1.98542 2.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.985417748196 = 2*cos(2*π/52) = 2*cos(π/26) 1.941883634852 = 2*cos(4*π/52) = 2*cos(π/13) 1.870032485371 = 2*cos(6*π/52) = 2*cos(3*π/26) 1.770912051306 = 2*cos(8*π/52) = 2*cos(2*π/13) 1.645967731787 = 2*cos(10*π/52) = 2*cos(5*π/26) 1.497021496342 = 2*cos(12*π/52) = 2*cos(3*π/13) 1.326245316482 = 2*cos(14*π/52) = 2*cos(7*π/26) 1.136129493462 = 2*cos(16*π/52) = 2*cos(4*π/13) 0.929446344088 = 2*cos(18*π/52) = 2*cos(9*π/26) 0.709209774085 = 2*cos(20*π/52) = 2*cos(5*π/13) 0.478631328575 = 2*cos(22*π/52) = 2*cos(11*π/26) 0.241073360511 = 2*cos(24*π/52) = 2*cos(6*π/13) Symmetry of Rotations and Cartesian products Ag R+d+g+i+k+m R_z, z², z⁴, z⁶ E1g R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} E2g d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3g g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4g g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5g i+k+m {xz(x²−(5+2√5)y²)(x²−(5−2√5)y²), yz((5+2√5)x²−y²)((5−2√5)x²−y²)} E6g i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7g k+m E8g k+m E9g m E10g m Au p+f+h+j+l z, z³, z⁵ E1u p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2u f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E3u f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4u h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E5u h+j+l {x(x²−(5+2√5)y²)(x²−(5−2√5)y²), y((5+2√5)x²−y²)((5−2√5)x²−y²)} E6u j+l E7u j+l E8u l E9u l Notes: α The order of the C_52h point group is 104, and the order of the principal axis (C₅₂) is 52. The group has 54 irreducible representations. β The C_52h point group is generated by two symmetry elements, which are canonically chosen as C₅₂ and i. Other possible choices are C₅₂ and σ_h, or less commonly S₅₂ with either i or σ_h. γ The lowest nonvanishing multipole moment in C_52h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_52h group is Abelian because all its symmetry operations are coaxial. This is a sufficient condition. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. ε Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 100 cases have been combined into 50 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 50 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 50 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. η Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. θ The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.

This Character Table for the C_52h point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.

	C_50h
	C_51h
C₅₂ C_52v	C_52h	D₅₂ D_52h D_52d S₅₂
	C_53h
	C_54h

Character table for the C52h point group

Character table for the C_52h point group